Illinois Journal of Mathematics
Volume 59, Number 2, Summer 2015, Pages 449–483
S 0019-2082
UNIQUE PSEUDO-EXPECTATIONS FOR C ∗ -INCLUSIONS
DAVID R. PITTS AND VREJ ZARIKIAN
Dedicated to E. G. Effros on the occasion of his 80th birthday
Abstract. Given an inclusion D ⊆ C of unital C ∗ -algebras (with
common unit), a unital completely positive linear map Φ of C
into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. Pseudo-expectations
are generalizations of conditional expectations, but with the advantage that they always exist. The set PsExp(C, D) of all
pseudo-expectations is a convex set, and when D is Abelian, we
prove a Krein–Milman type theorem showing that PsExp(C, D)
can be recovered from its set of extreme points. In general,
PsExp(C, D) is not a singleton. However, there are large and
natural classes of inclusions (e.g., when D is a regular MASA in
C) such that there is a unique pseudo-expectation. Uniqueness
of the pseudo-expectation typically implies interesting structural
properties for the inclusion. For general inclusions of C ∗ -algebras
with D Abelian, we give a characterization of the unique pseudoexpectation property in terms of order structure; and when C
is Abelian, we are able to give a topological description of the
unique pseudo-expectation property.
As applications, we show that if an inclusion D ⊆ C has a
unique pseudo-expectation Φ which is also faithful, then the
C ∗ -envelope of any operator space X with D ⊆ X ⊆ C is the C ∗ subalgebra of C generated by X ; we also show that for many interesting classes of C ∗ -inclusions, having a faithful unique pseudoexpectation implies that D norms C, although this is not true in
general.
Received August 12, 2015; received in final form December 23, 2015.
This work was partially supported by a grant from the Simons Foundation (#316952 to
David Pitts).
2010 Mathematics Subject Classification. Primary 46L05, 46L07, 46L10. Secondary
46M10.
c
2016
University of Illinois
449
450
D. R. PITTS AND V. ZARIKIAN
1. Introduction
The goal of this paper is to investigate the unique pseudo-expectation property for C ∗ -inclusions. A C ∗ -inclusion is a pair (C, D) of unital C ∗ -algebras
with D ⊆ C and which have the same unit. For any unital C ∗ -algebra D, there
exists an injective envelope I(D) for D [13]. That is, I(D) is an injective object in the category OSys1 of operator systems and unital completely positive
(ucp) maps, which contains D, and which is minimal with respect to these two
properties. In fact, I(D) is a C ∗ -algebra and D ⊆ I(D) is a C ∗ -subalgebra.
A pseudo-expectation is a ucp map Φ : C → I(D) which extends the identity
map on D.
Pseudo-expectations are natural generalizations of conditional expectations, and due to injectivity, have the distinct advantage that they are guaranteed to exist for any C ∗ -inclusion. Pseudo-expectations were introduced by
Pitts in [28] and were used there as a replacement for conditional expectations
in settings where no conditional expectation exists.
One significant difference between conditional expectations and pseudoexpectations arises when one attempts to iterate these maps. For a conditional expectation E : C → D, we have that E ◦ E = E (i.e., a conditional
expectation is an idempotent map). For a pseudo-expectation Φ : C → I(D),
the composition Φ ◦ Φ is typically undefined, since I(D) is usually not contained in C. This technical difficulty of pseudo-expectations is far outweighed
by the aforementioned benefit, that pseudo-expectations always exist for any
C ∗ -inclusion.
We view the uniqueness and faithfulness properties of pseudo-expectations
as giving a measure of the relative size of a subalgebra inside the containing
algebra. To orient the reader with this philosophy, we begin by explaining
how the unique pseudo-expectation property fits with the program of deciding
when a C ∗ -subalgebra is large/substantial/rich in its containing C ∗ -algebra.
1.1. Large subalgebras. Let (C, D) be a C ∗ -inclusion. There are many
ways of expressing that D is “large” (or “substantial”, or “rich”) in C. For
example:
(ARC) The relative commutant Dc = D ∩ C is Abelian.
(Reg) D is regular in C, meaning that span(N (C, D)) = C, where
N (C, D) = x ∈ C : xDx∗ ⊆ D, x∗ Dx ⊆ D
are the normalizers of D in C.
(Ess) D is essential in C, meaning that every nontrivial closed two-sided
ideal of C intersects D nontrivially.
(UEP) C has the unique extension property relative to D, meaning that
every pure state on D extends uniquely to a pure state on C.
UNIQUE PSEUDO-EXPECTATIONS
451
(Norming) D norms C, meaning that for all X ∈ Md×d (C),
X = sup RXC : R ∈ Ball M1×d (D) , C ∈ Ball Md×1 (D) .
Some of these conditions are purely algebraic, others purely analytic, and
yet others somewhere in between. Each of them has advantages and disadvantages, and their relative merits vary by context. Indeed, two desirable
properties for any condition which “measures” the largeness of D in C are the
following:
• Hereditary from above: If D is large in C and D ⊆ C0 ⊆ C is a C ∗ -algebra,
then D is large in C0 .
• Hereditary from below : If D is large in C and D ⊆ D0 ⊆ C is a C ∗ -algebra,
then D0 is large in C.
Table 1 shows which of these hereditary properties the various types of inclusions possess (an entry marked “?” indicates we do not know whether
the property holds). Only conditions (ARC) and (Norming) are known to the
authors to be both hereditary from above and below, see Table 1.
On the other hand, as the following example shows, condition (Ess) works
the best for the particular class of Abelian inclusions, in spite of its general
shortcomings.
Example 1.1. Suppose (A, D) = (C(Y ), C(X)) is an Abelian inclusion,
with corresponding continuous surjection j : Y → X. Then
• (A, D) always satisfies (ARC).
• (A, D) always satisfies (Reg).
• (A, D) satisfies (Ess) ⇐⇒ the only closed set K ⊆ Y such that j(K) = X
is Y itself.
• (A, D) satisfies (UEP) ⇐⇒ A = D.
• (A, D) always satisfies (Norming).
1.2. Unique expectations. If D is large in C, then there should not be
many ways to project C onto D. The most natural way to project a C ∗ -algebra
C onto a C ∗ -subalgebra D is via a conditional expectation. Recall that a conditional expectation for (C, D) is a ucp map E : C → D such that E|D = id.
Table 1. Hereditary properties of inclusions
Condition
ARC
Reg
Ess
UEP
Norming
Hereditary
from above
yes
?
no
yes
yes
Hereditary
from below
yes
no
yes
?
yes
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D. R. PITTS AND V. ZARIKIAN
A conditional expectation E : C → D is said to be faithful if E(x∗ x) = 0 implies
x = 0 (i.e., if E is faithful as a ucp map). Any convex combination of conditional expectations for (C, D) is again a conditional expectation for (C, D).
Thus a C ∗ -inclusion has either zero, one, or uncountably many conditional
expectations, and all three possibilities can occur.
In light of the previous discussion, it is reasonable to propose the following
property as yet another expression of the largeness of D in C:
(!CE) There is at most one conditional expectation E : C → D.
The utility of this property is seriously limited in two ways. First, for many
naturally arising C ∗ -inclusions, there are no conditional expectations at all;
and second, as the next two examples show, (!CE) fails to be hereditary from
above or below.
Example 1.2. Consider the C ∗ -inclusions
C[0, 1] ⊆ C [0, 1] × [0, 1] ⊆ B L2 [0, 1] × [0, 1] ,
where the first inclusion corresponds to the continuous surjection
j : [0, 1] × [0, 1] → [0, 1] : (s, t) → s.
Then there are no conditional expectations for the inclusion (B(L2 ([0, 1] ×
[0, 1])), C[0, 1]), since C[0, 1] is not injective (in the category OSys1 ). But
there are infinitely many conditional expectations for the inclusion (C([0, 1] ×
[0, 1]), C[0, 1]). Indeed,
Et : C [0, 1] × [0, 1] → C[0, 1] : g → g(·, t)
is a conditional expectation for each t ∈ [0, 1]. Thus, (!CE) is not hereditary
from above.
Example 1.3. Likewise, consider the C ∗ -inclusions
C[0, 1] ⊆ L∞ [0, 1] ⊆ B L2 [0, 1] .
There are no conditional expectations for the inclusion (B(L2 [0, 1]), C[0, 1]),
but infinitely many conditional expectations for (B(L2 [0, 1]), L∞ [0, 1]), see
[20]. Thus, (!CE) is not hereditary from below.
1.3. Unique pseudo-expectations. Recall that a ucp map Φ : C → I(D)
is a pseudo-expectation if it extends the inclusion of D into I(D). Clearly
every conditional expectation for (C, D) is a pseudo-expectation for (C, D), so
pseudo-expectations generalize conditional expectations. But pseudo-expectations always exist for any C ∗ -inclusion.
With the discussion of the previous section in mind, we are led to replace
condition (!CE) there by the following stronger condition:
(!PsE) There exists a unique pseudo-expectation for (C, D).
Or perhaps by the even stronger condition:
(f!PsE) There exists a unique pseudo-expectation for (C, D), which is faithful.
UNIQUE PSEUDO-EXPECTATIONS
453
We will see shortly that both of these conditions are hereditary from above
(Proposition 2.6). Compelling evidence that (!PsE) and (f!PsE) are closely
related to the largeness of D in C is provided by a striking result from [28]:
Theorem 1.4 (Pitts). Let (C, D) be a regular inclusion with D a MASA
in C.
(i) Then there exists a unique pseudo-expectation Φ : C → I(D).
(ii) If LΦ = {x ∈ C : Φ(x∗ x) = 0} is the left kernel of Φ, then LΦ is the unique
maximal D-disjoint ideal in C.
(iii) If Φ is faithful (i.e., if LΦ = 0), then D norms C.
Rephrasing Theorem 1.4 using the notation of this section, statement (i)
says that for a C ∗ -inclusion (C, D), with D maximal Abelian,
(Reg) =⇒ (!PsE).
Statements (ii) and (iii) imply that under the same hypotheses,
(Reg) ∧ (f!PsE) =⇒ (Ess) ∧ (Norming).
This paper is a systematic attempt to generalize Theorem 1.4. We characterize the unique pseudo-expectation property for various important classes
of C ∗ -inclusions, and we relate the unique pseudo-expectation property for a
C ∗ -inclusion (C, D) to other measures of the largeness of D in C, in particular conditions (ARC), (Reg), (Ess), (UEP), and (Norming) above. Necessarily,
we significantly develop the general theory of pseudo-expectations along the
way.
2. The unique pseudo-expectation property
2.1. Definitions and basic properties. In this section, we formally define pseudo-expectations and explore their basic properties. Before doing so,
we remind the reader of a few facts about injective envelopes and establish some standing assumptions used throughout the paper. All C ∗ -algebras
are assumed unital, and homomorphisms between C ∗ -algebras will always be
∗-homomorphisms which preserve the units. We will denote by OSys1 the
category whose objects are operator systems and whose morphisms are ucp
(unital completely positive) maps. A C ∗ -algebra is injective if it is injective
when viewed as an object in OSys1 . Let AbC∗ be the category of Abelian
C ∗ -algebras and homomorphisms. Clearly every object in AbC∗ is also an
object in OSys1 . An important observation found in [14] and [12] is that
an Abelian C ∗ -algebrais injective in AbC∗ if and only if it is injective in
OSys1 .
Theorem 2.1 (See [10] or [27]). Let D be a unital C ∗ -algebra. Then there
exists a unital C ∗ -algebra A and a unital ∗-monomorphism ι : D → A with the
following properties:
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D. R. PITTS AND V. ZARIKIAN
(i) A is injective;
(ii) if S is an injective object in OSys1 and τ : D → S is a unital complete
isometry, then there exists a unital complete isometry τ1 : A → S such
that τ = τ1 ◦ ι.
The pair (A, ι) is called an injective envelope for D, and it is “nearly”
unique. The ambiguity arises from the fact that in general, the choice of
τ1 in Theorem 2.1 is not unique. However, in the sequel, we will assume
that for a given C ∗ -algebra D under discussion, a choice of injective envelope
(I(D), ι) has been made. Furthermore, we will regard ι as an inclusion map
and suppress writing it. Thus, we will always regard D as a C ∗ -subalgebra of
I(D).
Definition 2.2. A pseudo-expectation for the C ∗ -inclusion (C, D) is a ucp
map Φ : C → I(D) such that Φ|D = id. We denote by PsExp(C, D) the collection of all pseudo-expectations for (C, D).
Proposition 2.3. Let (C, D) be a C ∗ -inclusion, Φ ∈ PsExp(C, D), and
LΦ = x ∈ C : Φ x∗ x = 0
be the left kernel of Φ. Then the following statements hold:
(i) Φ is a D-bimodule map. That is, Φ(d1 xd2 ) = d1 Φ(x)d2 for all x ∈ C,
d1 , d2 ∈ D.
(ii) LΦ is a closed left ideal in C which intersects D trivially. Furthermore,
LΦ is a right D-module.
Proof. The first statement follows from Choi’s lemma ([27, Corollary 3.19]);
the second is straightforward.
Proposition 2.4. Let (C, D) be a C ∗ -inclusion.
(i) The family PsExp(C, D) of all pseudo-expectations for (C, D) forms a
nonempty convex subset of UCP(C, I(D)), the ucp maps from C into I(D).
In fact, PsExp(C, D) is a face of UCP(C, I(D)). Thus, any extreme point
of PsExp(C, D) is an extreme point of UCP(C, I(D)).
(ii) If CE(C, D) denotes the collection of all conditional expectations for
(C, D), then CE(C, D) ⊆ PsExp(C, D). Of course it can happen that
CE(C, D) = ∅, whereas PsExp(C, D) = ∅, by injectivity.
Proof. We only prove that PsExp(C, D) is a face of UCP(C, I(D)). Indeed,
suppose Φ ∈ PsExp(C, D) and Φ = λΦ1 + (1 − λ)Φ2 , where Φ1 , Φ2 belong to
UCP(C, I(D)) and λ ∈ (0, 1). For any u ∈ U (D) (the unitary group of D), we
have that
u = Φ(u) = λΦ1 (u) + (1 − λ)Φ2 (u).
Since Φ1 (u), Φ2 (u) ∈ Ball(I(D)) and
u ∈ U (D) ⊆ U I(D) ⊆ Ext Ball I(D) ,
UNIQUE PSEUDO-EXPECTATIONS
455
we conclude that Φ1 (u) = Φ2 (u) = u. It follows that Φ1 (d) = Φ2 (d) = d for all
d ∈ D, so that Φ1 , Φ2 ∈ PsExp(C, D).
Definition 2.5. We say that a C ∗ -inclusion (C, D) has the unique pseudoexpectation property (!PsE) if there exists a unique Φ ∈ PsExp(C, D). If, in
addition, Φ is faithful, then we say that (C, D) has the faithful unique pseudoexpectation property (f!PsE).
As in the Introduction, we say that a property of C ∗ -inclusions is hereditary
from above if whenever (C, D) has the property and D ⊆ C0 ⊆ C is a C ∗ -algebra,
then (C0 , D) has the property.
Proposition 2.6. The unique pseudo-expectation property is hereditary
from above, as is the faithful unique pseudo-expectation property.
Proof. Suppose PsExp(C, D) = {Φ}. Let D ⊆ C0 ⊆ C be a C ∗ -algebra, and
fix θ ∈ PsExp(C0 , D). By injectivity, there exists a ucp map Θ : C → I(D) such
that Θ|C0 = θ. Then Θ|D = θ|D = id, so that Θ ∈ PsExp(C, D). It follows that
Θ = Φ, which implies θ = Θ|C0 = Φ|C0 . Thus PsExp(C0 , D) = {Φ|C0 }. If Φ is
faithful, then so is Φ|C0 .
On the other hand, if (C, D) has the unique pseudo-expectation property and D ⊆ D0 ⊆ C is a C ∗ -algebra, then (C, D0 ) may not have the unique
pseudo-expectation property (see Example 4.1). That is, the unique pseudoexpectation property is not hereditary from below.
2.2. Elementary examples. In this section, we give some examples of C ∗ inclusions with (and without) the unique pseudo-expectation property. These
examples are “elementary”, insofar as we can prove that they are actually
examples without any additional technology. Later, after we have developed
some general theory for pseudo-expectations, we will give a number of “advanced” examples.
Example 2.7 (Regular MASA inclusions). Let (C, D) be a regular MASA
inclusion. Then (C, D) has the unique pseudo-expectation property, by Pitts’
Theorem 1.4. Two classes of regular MASA inclusions which appear in the literature are C ∗ -diagonals in the sense of Kumjian [22], and Cartan subalgebras
in the sense of Renault [32].
Example 2.8 (Atomic MASA). The inclusion (B(2 ), ∞ ) has the faithful
unique pseudo-expectation property. Indeed, ∞ is injective (since it is an
Abelian W ∗ -algebra) and there exists a unique conditional expectation E :
B(2 ) → ∞ , which is faithful [20, Theorem 1].
Example 2.9 (Diffuse MASA). The inclusion (B(L2 [0, 1]), L∞ [0, 1]) has
infinitely many pseudo-expectations, none of which are faithful. However, the
inclusion, (L∞ [0, 1] + K(L2 [0, 1]), L∞ [0, 1]) has a unique pseudo-expectation,
which is not faithful.
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D. R. PITTS AND V. ZARIKIAN
Proof. Since L∞ [0, 1] is injective, conditional expectations and pseudoexpectations for (B(L2 [0, 1]), L∞ [0, 1]) coincide. By Theorem 2 and Remark 5
of [20], there are infinitely many conditional expectations B(L2 [0, 1]) →
L∞ [0, 1], all of which annihilate K(L2 [0, 1]). Now suppose E : L∞ [0, 1] +
K(L2 [0, 1]) → L∞ [0, 1] is a conditional expectation. Then E extends to a
conditional expectation Ẽ : B(L2 [0, 1]) → L∞ [0, 1]. Thus, by the previous discussion,
E(d + h) = Ẽ(d + h) = d
∞
for all d ∈ L [0, 1], h ∈ K(L2 [0, 1]).
Remark 2.10. Let (C, D) be a C ∗ -inclusion. Then we have C ∗ -inclusions
D ⊆ C ⊆ I(C). By Theorem 2.1, it follows that we have an operator system inclusion I(D) ⊆ I(C). If D is Abelian, then in fact we have a C ∗ -inclusion
I(D) ⊆ I(C) [12, Thm. 2.21]. In that case, if Φ : C → I(D) is a pseudoexpectation for (C, D), then it is not hard to see that any ucp extension
Φ̃ : I(C) → I(D) of Φ is a conditional expectation for (I(C), I(D)). As the
previous example shows, this extension need not be unique. Indeed, by [14,
Ex. 5.3], I(L∞ [0, 1] + K(L2 [0, 1])) = B(L2 [0, 1]).
Next, we consider C ∗ -inclusions (C, D) such that there is a monomorphism
of C into I(D). By [13, Lemma 4.6], these are precisely the operator space
essential inclusions. A C ∗ -inclusion (C, D) is operator space essential (OSE)
if every complete contraction u : C → B(H) which is completely isometric on
D is actually completely isometric on C.
Example 2.11 (OSE inclusions). Let D be an arbitrary unital C ∗ -algebra.
Then (I(D), D) has the faithful unique pseudo-expectation property. More
generally, if D ⊆ C ⊆ I(D) are C ∗ -inclusions, then (C, D) has the faithful
unique pseudo-expectation property, by Proposition 2.6.
Proof. Let Φ ∈ PsExp(I(D), D). Then Φ : I(D) → I(D) is a ucp map such
that Φ|D = id. By the rigidity of the injective envelope, Φ = id.
Example 2.12 (UEP MASA inclusions). Let (C, D) be a C ∗ -inclusion, with
D Abelian. Assume that (C, D) has the unique extension property (UEP),
meaning that every pure state on D extends uniquely to a pure state on C.
(This forces D to be a MASA in C.) Then (C, D) has the unique pseudoexpectation property. In fact, the unique pseudo-expectation is a conditional
expectation.
Proof. By [3, Cor. 2.7], we have the direct sum decomposition
C = D + span [C, D] .
If Φ ∈ PsExp(C, D), then by Proposition 2.3 and the fact that I(D) is Abelian,
Φ(xd − dx) = Φ(xd) − Φ(dx) = Φ(x)d − dΦ(x) = 0,
The result follows.
x ∈ C, d ∈ D.
UNIQUE PSEUDO-EXPECTATIONS
457
Remark 2.13. Initially the study of UEP inclusions (C, D) focused on the
case D Abelian, and there has been substantial work in this direction. Later
work has made progress in the general setting [6]. It would be interesting
to know whether a general UEP inclusion (C, D) has the unique pseudoexpectation property. The inclusion (Cr∗ (Fm ), Cr∗ (Fn )) for m > n ≥ 2, may
provide a possible test case [2, Thm. 2.6].
Example 2.14. Let M be a II1 factor with separable predual and D ⊆ M
be a MASA. More generally, let M be any II1 factor and D ⊆ M be a singlygenerated MASA. Then (M, D) does not have the unique pseudo-expectation
property [1, Thm. 4.4].
3. Some general theory
In this section, we prove some general results about pseudo-expectations,
which we will use later to analyze more complicated examples than those
considered so far.
3.1. Left kernel. Let (C, D) be a C ∗ -inclusion. We say that a closed twosided ideal J C is D-disjoint if D ∩ J = 0. It is not hard to prove that every
D-disjoint ideal of C is contained in a maximal D-disjoint ideal of C.
As seen in Theorem 1.4, if (C, D) is a regular MASA inclusion, then there
exists a unique maximal D-disjoint ideal in C, namely the left kernel LΦ of the
unique pseudo-expectation Φ : C → I(D). In general, for a C ∗ -inclusion (C, D)
with unique pseudo-expectation Φ, the left kernel LΦ is only a left ideal of
C, rather than a two-sided ideal (see Example 3.10 below). Nevertheless, we
have the following structural result for general C ∗ -inclusions with the unique
pseudo-expectation property.
Proposition 3.1. Let (C, D) be a C ∗ -inclusion. If (C, D) has unique
pseudo-expectation Φ, then there exists a unique maximal D-disjoint ideal
I C. Furthermore, I ⊆ LΦ , the left kernel of Φ.
Proof. Let J C be a D-disjoint ideal. Then the map D + J → D : d + h →
d is a unital ∗-homomorphism, which extends by injectivity to a pseudoexpectation for (C, D), necessarily Φ. Thus J ⊆ ker(Φ). If h ∈ J , then h∗ h ∈
J , which implies Φ(h∗ h) = 0, which in turn implies h ∈ LΦ . Thus J ⊆ LΦ . It
follows that
{J : J C, D ∩ J = 0} ⊆ LΦ ,
and so
I = span
{J : J C, D ∩ J = 0} ⊆ LΦ .
Thus I is the unique maximal D-disjoint ideal of C.
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D. R. PITTS AND V. ZARIKIAN
3.2. Characterization: Every pseudo-expectation is faithful. In this
section, we characterize the property “every pseudo-expectation is faithful”
for arbitrary C ∗ -inclusions (C, D) in terms of the (hereditary) D-disjoint
ideal structure of C (Theorem 3.5). Formally, the property “every pseudoexpectation is faithful” is weaker than the faithful unique pseudo-expectation
property. On the other hand, we have no examples showing that it is strictly
weaker. So in principle, Theorem 3.5 could be a characterization of the faithful
unique pseudo-expectation property. We list this as an open problem.
Question 3.2. Does the property “every pseudo-expectation is faithful”
imply the faithful unique pseudo-expectation property?
To proceed with our characterization, we will need two notions from earlier
in the paper. First, recall that a closed two-sided ideal J C is D-disjoint if
D ∩ J = 0. Second, recall that a C ∗ -inclusion (C, D) is essential (Ess) if every
nontrivial closed two-sided ideal of C intersects D nontrivially. The following
proposition relates these two notions with each other, as well as to a useful
mapping property.
Proposition 3.3. Let (C, D) be a C ∗ -inclusion. Then the following are
equivalent:
(i) (C, D) is essential.
(ii) The only D-disjoint ideal of C is 0.
(iii) Whenever π : C → B(H) is a unital ∗-homomorphism such that π|D is
faithful, then π itself is faithful.
Proof. (i) ⇐⇒ (ii) Tautological.
(ii) =⇒ (iii) Suppose the only D-disjoint ideal of C is the trivial ideal. Let
π : C → B(H) be a unital ∗-homomorphism such that π|D is faithful. Then
ker(π) C is a D-disjoint ideal. By assumption, ker(π) = 0, so π is faithful.
(iii) =⇒ (ii) Conversely, suppose that for every unital ∗-homomorphism
π : C → B(H), π is faithful whenever π|D is faithful. Let J C be a D-disjoint
ideal. Then q : C → C/J : x → x + J is a unital ∗-homomorphism such that
q|D is faithful. By assumption, q is faithful, so J = 0.
As we saw in the Introduction, the condition (Ess) is not hereditary from
above. Indeed, (M2×2 (C), CI) satisfies (Ess), since M2×2 (C) is simple, and
CI ⊆ C ⊕ C ⊆ M2×2 (C) is a C ∗ -algebra, but (C ⊕ C, CI) fails (Ess). To resolve
this issue, we introduce the following stronger condition:
Definition 3.4. We say that a C ∗ -inclusion (C, D) is hereditarily essential
if (C0 , D) is essential whenever D ⊆ C0 ⊆ C is a C ∗ -algebra.
Now comes the promised characterization.
Theorem 3.5. Let (C, D) be a C ∗ -inclusion. Then the following are equivalent:
UNIQUE PSEUDO-EXPECTATIONS
459
(i) Every pseudo-expectation Φ ∈ PsExp(C, D) is faithful.
(ii) (C, D) is hereditarily essential.
Proof. (i) =⇒ (ii) Suppose that every pseudo-expectation Φ ∈ PsExp(C, D)
is faithful. Let D ⊆ C0 ⊆ C be a C ∗ -algebra and J0 C0 be a D-disjoint ideal.
Then Φ0 : D + J0 → D : d + h → d is a unital ∗-homomorphism. By injectivity,
there exists Φ ∈ PsExp(C, D) such that such that Φ|D+J0 = Φ0 . Since Φ is
faithful, so is Φ0 , which implies J0 = 0. It follows that (C0 , D) is essential,
which implies (C, D) is hereditarily essential.
(ii) =⇒ (i) Conversely, suppose that (C, D) is hereditarily essential. Let
Φ ∈ PsExp(C, D) and x ∈ LΦ (the left kernel of Φ). Define C0 = C ∗ (D, |x|),
so that D ⊆ C0 ⊆ C, and let J0 C0 be the closed two-sided ideal generated
by |x|. We claim that J0 ⊆ LΦ . Indeed, J0 = span{w|x|d : w ∈ C0 , d ∈ D}
and LΦ is both a closed left ideal and a right D-module in C containing |x|
(Proposition 2.3). Since D ∩ LΦ = 0, D ∩ J0 = 0, and since (C0 , D) is essential
by assumption, J0 = 0. Thus |x| = 0, which implies x = 0. Hence LΦ = 0, so
Φ is faithful.
3.3. Quotients. We next examine the behavior of the unique pseudo-expectation property with respect to quotients. Specifically, for a closed two-sided
ideal J C, we are interested to know when the unique pseudo-expectation
property for (C, D) passes to (C/J , D/(J ∩ D)). If J ∩ D = 0, then the answer
is “always”, and faithfulness is preserved.
Proposition 3.6. Let (C, D) be a C ∗ -inclusion and J C be a D-disjoint
ideal. If (C, D) has the unique pseudo-expectation property, then so does
(C/J , D). If (C, D) has the faithful unique pseudo-expectation property, then
so does (C/J , D) (trivially, because J = 0).
Proof. Suppose PsExp(C, D) = {Φ}. Let θ ∈ PsExp(C/J , D). Then θ ◦ q ∈
PsExp(C, D), where q : C → C/J is the quotient map. Thus θ ◦ q = Φ, which
implies θ(x + J ) = Φ(x), x ∈ C. Hence, PsExp(C/J , D) = {θ}. If Φ is faithful,
then J = 0, by Theorem 3.5.
Remark 3.7. If D ∩ J = 0, then it is entirely possible that (C, D) has a
unique pseudo-expectation but (C/J , D/(J ∩ D)) does not (see Example 4.2).
In order to obtain a positive result when D ∩ J = 0, we require J ∩ D D
to be regular. Recall that if A is a unital C ∗ -algebra and I A, then
I ⊥ = {a ∈ A : aI = Ia = 0} A.
Also, I is regular if I ⊥⊥ = (I ⊥ )⊥ = I. Combining [16, Lemma 1.3(iii)] with
[15, Theorem 6.3], one finds that given a regular ideal I A, there exists a
unique projection p ∈ Z(I(A)) such that I = {a ∈ A : ap = a}. In that case,
the unital ∗-isomorphism A/I → Ap⊥ : a + I → ap⊥ extends uniquely to a
unital ∗-isomorphism I(A/I) ∼
= I(A)p⊥ .
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D. R. PITTS AND V. ZARIKIAN
Theorem 3.8. Let (C, D) be a C ∗ -inclusion and J C. If (C, D) has
the unique pseudo-expectation property and J ∩ D D is regular, then the
inclusion (C/J , D/(J ∩ D)) has the unique pseudo-expectation property.
Proof. Let p ∈ Z(I(D)) be the unique projection such that J ∩ D = {d ∈ D :
dp = d}. Then the unital ∗-isomorphism D/(J ∩ D) → Dp⊥ : d + (J ∩ D) →
dp⊥ extends uniquely to a unital ∗-isomorphism I(D/(J ∩ D)) ∼
= I(D)p⊥ .
Now suppose PsExp(C, D) = {Φ} and let θ ∈ PsExp(C/J , D/(J ∩ D)). Then
θ : C/J → I(D)p⊥ is a ucp map such that θ(d + J ) = dp⊥ , d ∈ D. Define
Θ : C → I(D) by
Θ(x) = θ(x + J ) + Φ(x)p, x ∈ C.
Then Θ ∈ PsExp(C, D), which implies Θ = Φ, which in turn implies
θ(x + J ) = Φ(x)p⊥ ,
x ∈ C.
Thus (C/J , D/(J ∩ D)) has the unique pseudo-expectation property.
Remark 3.9. If (C, D) has the faithful unique pseudo-expectation property
and J C, then (C/J , D/(J ∩ D)) need not have the faithful unique pseudoexpectation property, even if J ∩ D D is regular (see Example 4.3).
A very interesting example not covered by the results of this section occurs
when C = B(2 ), D = ∞ , and J = K(2 ), so that
C/J , D/(J ∩ D) = B 2 /K 2 , ∞ /c0 .
Indeed, c0 ∞ is not regular, since c⊥⊥
= ∞ . Our analysis of this exam0
ple is greatly simplified by the recent remarkable affirmative solution to the
Kadison–Singer problem [26].
Example 3.10 (Calkin algebra). The inclusion (B(2 )/K(2 ), ∞ /c0 )
has the unique pseudo-expectation property. In fact, the unique pseudoexpectation is a conditional expectation which is not faithful.
Proof. By [26], the inclusion (B(2 ), ∞ ) has the unique extension property
(UEP). By [3, Lemma 3.1], (B(2 )/K(2 ), ∞ /c0 ) has (UEP) as well. Thus,
(B(2 )/K(2 ), ∞ /c0 ) has a unique pseudo-expectation Ẽ, which is actually a
conditional expectation, by Example 2.12. In fact,
Ẽ x + K 2 = E(x) + c0 , x ∈ B 2 ,
where E : B(2 ) → ∞ is the unique conditional expectation. Letting h ∈
B(2 )+ be the Hilbert matrix [7], we see that Ẽ(h + K(2 )) = 0, but h +
K(2 ) = 0.
Remark 3.11. Example 3.10 furnishes an instance of a C ∗ -inclusion (C, D)
with a unique pseudo-expectation Φ, such that LΦ is not a two-sided ideal of C.
Indeed, C is simple but LΦ = 0 in Example 3.10. This should be compared
with Theorem 1.4.
UNIQUE PSEUDO-EXPECTATIONS
461
3.4. Abelian relative commutant. As mentioned in the Introduction, the
unique pseudo-expectation property for a C ∗ -inclusion (C, D) can be thought
of as an expression of the fact that D is “large” in C. A more familiar algebraic expression of the largeness of D in C is that Dc = D ∩ C, the relative
commutant of D in C, is “small” (Abelian). In Corollary 3.14 below, we show
that the faithful unique pseudo-expectation property implies that the relative
commutant is Abelian, symbolically
(f!PsE) =⇒ (ARC).
We expect that the hypothesis of faithfulness is not needed for this result, but
we have not been able to eliminate it.
Theorem 3.12. Let (C, D) be a C ∗ -inclusion. Assume that there exists a
faithful pseudo-expectation Φ ∈ PsExp(C, D). If Dc is not Abelian, then there
exist infinitely many pseudo-expectations for (C, D), some of which are not
faithful.
Proof. We may assume that C ⊆ B(H) for some Hilbert space H. If Dc is
not Abelian, then there exists x ∈ Dc with x = 1 and x2 = 0 ([9, p. 288]).
Let x = u|x| be the polar decomposition, so that u ∈ D is a partial isometry
with initial space ran(|x|) and final space ran(x). Since
⊥
ran(x) ⊆ ker(x) = ker |x| = ran |x| ,
we find that u2 = 0. It follows that
p1 = u∗ u,
p2 = uu∗ ,
and
p3 = 1 − u∗ u − uu∗
are orthogonal projections in D . For λ ∈ [0, 1] define θλ : B(H) → B(H) by
θλ (t) = λp1 tp1 + (1 − λ)u∗ tu + λutu∗ + (1 − λ)p2 tp2 + p3 tp3 .
Then θλ is a ucp map such that
θλ |D = id,
θλ x∗ x = λ x∗ x+xx∗ ,
and
θλ xx∗ = (1−λ) x∗ x+xx∗ .
Consider the operator system
S := D + Cx∗ x + Cxx∗ ⊆ C.
Since θλ (S) ⊆ S,
Φ0λ := Φ ◦ θλ |S : S → I(D)
is a well-defined ucp map such that
Φ0λ |D = id,
Φ0λ x∗ x = λΦ x∗ x + xx∗ ,
Φ0λ xx∗ = (1 − λ)Φ x∗ x + xx∗ .
and
By injectivity, there exists Φλ ∈ PsExp(C, D) such that Φλ |S = Φ0λ . Since
Φ is faithful, Φ(x∗ x + xx∗ ) = 0, and so Φλ = Φμ if λ = μ. Consequently,
{Φλ : λ ∈ [0, 1]} is an infinite family of pseudo-expectations for (C, D), some
of which are not faithful (namely Φ0 and Φ1 ).
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D. R. PITTS AND V. ZARIKIAN
Remark 3.13. In Theorem 3.12, we may remove the hypothesis that there
exists a faithful pseudo-expectation Φ ∈ PsExp(C, D), provided we strengthen
the hypothesis on Dc . For example, we could ask that Dc contain a halving
projection. In that case, the proof simplifies substantially.
Corollary 3.14. Let (C, D) be a C ∗ -inclusion. If (C, D) has the faithful
unique pseudo-expectation property, then Dc is Abelian.
Remark 3.15. As indicated earlier, we expect Corollary 3.14 to remain
true without the assumption of faithfulness. At this point, however, we do
not have a proof, even in the case D Abelian. On the other hand, the case of
W ∗ -inclusions is completely settled in the affirmative (Corollary 5.3).
3.5. Characterization: Unique pseudo-expectation property for
Abelian subalgebras. In this section, we give an order-theoretic characterization of the unique pseudo-expectation property for C ∗ -inclusions (C, D),
with D Abelian. We remind the reader that if D is a unital Abelian C ∗ algebra, then I(D) is order complete, meaning that every nonempty set
S ⊆ I(D)sa with an upper bound has a supremum [36, Prop. III.1.7].
Theorem 3.16. Let (C, D) be a C ∗ -inclusion, with D Abelian. Then the
following are equivalent:
(i) (C, D) has the unique pseudo-expectation property.
(ii) For all x ∈ Csa ,
sup {d ∈ Dsa : d ≤ x} = inf {d ∈ Dsa : d ≥ x}.
I(D)
I(D)
Proof. For x ∈ Csa , set
(x) = sup {d ∈ Dsa : d ≤ x}
I(D)
and
u(x) = inf {d ∈ Dsa : d ≥ x}.
I(D)
It is easy to see that (x) ≤ u(x). (If e ∈ Dsa and e ≥ x, then d ≤ e for all
d ∈ Dsa such that d ≤ x. Thus, (x) ≤ e. Since the choice of e was arbitrary,
(x) ≤ u(x).) Clearly, (d) = d = u(d) for all d ∈ Dsa .
(i) =⇒ (ii) Let x ∈ Csa \ Dsa and suppose a ∈ I(D)sa satisfies (x) ≤ a ≤
u(x). Since D ∩ Cx = 0,
Φ0 : D + Cx → I(D) : d + λx → d + λa
is a well-defined linear map such that Φ0 |D = id. Suppose d + λx ≥ 0, so that
d ∈ Dsa and λ ∈ R.
Case 1: If λ = 0, then d ≥ 0, which implies d + λa = d ≥ 0.
Case 2: If λ > 0, then x ≥ − λ1 d, which implies − λ1 d ≤ (x) ≤ a, which in turn
implies d + λa ≥ 0.
Case 3: If λ < 0, then x ≤ − λ1 d, which implies a ≤ u(x) ≤ − λ1 d, which in turn
implies d + λa ≥ 0.
UNIQUE PSEUDO-EXPECTATIONS
463
The preceding analysis shows that Φ0 is positive, and since I(D) is Abelian, it
is actually completely positive. By injectivity, there exists a ucp map Φ : C →
I(D) such that Φ|D+Cx = Φ0 . Then Φ is a pseudo-expectation for (C, D) such
that Φ(x) = a. It follows that if there exists x ∈ Csa such that (x) = u(x),
then (C, D) admits multiple pseudo-expectations.
(ii) =⇒ (i) Conversely, suppose Φ ∈ PsExp(C, D). Let x ∈ Csa . If d ∈ Dsa
and d ≤ x, then d = Φ(d) ≤ Φ(x), which implies (x) ≤ Φ(x). Likewise if
d ∈ Dsa and d ≥ x, then d = Φ(d) ≥ Φ(x), which implies u(x) ≥ Φ(x). Thus
if (x) = u(x) for all x ∈ Csa , then Φ is uniquely determined on Csa , therefore
on C.
3.6. A Krein–Milman theorem for pseudo-expectations when the
subalgebra is Abelian. The purpose of this section is to prove a Krein–
Milman theorem for the pseudo-expectation space PsExp(C, D), valid for C ∗ inclusions (C, D), with D Abelian. Our goal is to show that there is a rich
supply of extreme points in PsExp(C, D). It will then follow that uniqueness of pseudo-expectations is equivalent to uniqueness of extreme pseudoexpectations. One approach to this type of result might be the following: first,
introduce an appropriate locally convex topology on the set of all bounded
linear maps from C into I(D); second, show that PsExp(C, D) is compact in
this topology; and finally, apply the usual Krein–Milman theorem. While this
may be a viable approach, it is not clear (at least to us) how to define such a
topology, so we proceed instead using a route through convexity theory, which
is perhaps less well-traveled.
Our key tool is Kutateladze’s Krein–Milman theorem for subdifferentials of
sublinear operators into Kantorovich spaces [23]. Let V and W be real vector
spaces. Assume further that W is a Kantorovich space, meaning that W is
a vector lattice such that every nonempty subset with an upper bound has a
supremum. Suppose Q : V → W a sublinear operator, meaning that
• Q(αv) = αQ(v) for all v ∈ V , α ≥ 0;
• Q(v1 + v2 ) ≤ Q(v1 ) + Q(v2 ) for all v1 , v2 ∈ V .
Let ∂Q be the subdifferential of Q:
∂Q = T ∈ Lin(V, W ) : T (v) ≤ Q(v), v ∈ V .
(Here Lin(V, W ) denotes the set of all real linear maps from V to W .) Kutateladze’s version of the Krein–Milman theorem is the following.
Theorem 3.17 (Kutateladze [23]). Let V and W be real vector spaces with
W a Kantorovich space, and suppose Q : V → W is a sublinear operator. Then
the following statements hold:
(i) Ext(∂Q) = ∅.
(ii) For v ∈ V , define P (v) = supW {T (v) : T ∈ Ext(∂Q)}. Then P : V → W
is a sublinear operator and ∂Q = ∂P .
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D. R. PITTS AND V. ZARIKIAN
We are now ready to apply Kutateladze’s theorem to our setting.
Theorem 3.18. Let (C, D) be a C ∗ -inclusion, with D Abelian. Then the
following statements hold:
(i) Ext(PsExp(C, D)) = ∅.
(ii) For x ∈ Csa , define P (x) = supI(D) {Ψ(x) : Ψ ∈ Ext(PsExp(C, D))}. Then
PsExp(C, D) = Φ ∈ UCP C, I(D) : Φ(x) ≤ P (x) for all x ∈ Csa .
In particular,
∃!Φ ∈ PsExp(C, D) ⇐⇒ ∃!Ψ ∈ Ext PsExp(C, D) .
Proof. Since D is Abelian, I(D)sa is a Kantorovich space. For all x ∈ Csa ,
define
Q(x) = sup Φ(x) : Φ ∈ PsExp(C, D) .
I(D)
It is easy to see that Q : Csa → I(D)sa is a sublinear operator. We claim that
PsExp(C, D) = Φ ∈ UCP C, I(D) : Φ|Csa ∈ ∂Q = {T̃ : T ∈ ∂Q},
where T̃ : C → I(D) is the complexification of T : Csa → I(D)sa . Indeed, the
inclusions of the first set into the second, and the second set into the third,
are tautological. Now let T ∈ ∂Q. Then
x ∈ C+
=⇒
−T (x) = T (−x) ≤ Q(−x) ≤ 0
=⇒
T (x) ≥ 0.
Thus T̃ is positive, and since I(D) is Abelian, completely positive. Also
d ∈ Dsa
=⇒
±T (d) = T (±d) ≤ Q(±d) = ±d
=⇒
T (d) = d.
Therefore, T̃ ∈ PsExp(C, D).
Invoking Kutateladze’s Krein–Milman theorem, we have that
Ext PsExp(C, D) = ∅,
and in fact
PsExp(C, D) = Φ ∈ UCP C, I(D) : Φ|Csa ∈ ∂P ,
where for all x ∈ Csa ,
P (x) = sup T̃ (x) : T ∈ Ext(∂Q) = sup Ψ(x) : Ψ ∈ Ext PsExp(C, D) .
I(D)
I(D)
If there exists a unique Φ ∈ PsExp(C, D), then clearly there exists a
unique Ψ ∈ Ext(PsExp(C, D)). Conversely, suppose there exists a unique
Ψ ∈ Ext(PsExp(C, D)). Then for all Φ ∈ PsExp(C, D),
x ∈ Csa
and so Φ = Ψ.
=⇒
±Φ(x) = Φ(±x) ≤ P (±x) = Ψ(±x) = ±Ψ(x)
=⇒
Φ(x) = Ψ(x),
UNIQUE PSEUDO-EXPECTATIONS
465
3.7. Abelian inclusions. In this section, we consider the unique pseudoexpectation property for C ∗ -inclusions (A, D), with A Abelian. By Gelfand
duality, these are precisely the C ∗ -inclusions (C(Y ), C(X)), where X and
Y are compact Hausdorff spaces. We recall that unital ∗-monomorphisms
π : C(X) → C(Y ) correspond bijectively to continuous surjections j : Y → X.
Indeed, if j : Y → X is a continuous surjection, then πj : C(X) → C(Y ) :
f → f ◦ j is a unital ∗-monomorphism. We may identify πj (C(X)) with the
continuous functions on Y which are constant on the fibers j −1 (x), x ∈ X.
Conversely, if π : C(X) → C(Y ) is a unital ∗-monomorphism, then for each
y ∈ Y there exists a unique j(y) ∈ X such that δy ◦ π = δj(y) , and it is easy to
verify that j : Y → X is a continuous surjection such that πj = π.
In light of Theorem 3.18, to characterize when PsExp(A, D) is a singleton,
it suffices to characterize when Ext(PsExp(A, D)) is a singleton. As we saw
in Proposition 2.4,
Ext PsExp(A, D) ⊆ Ext UCP A, I(D) .
On the other hand, Ψ ∈ Ext(UCP(A, I(D))) iff Ψ is multiplicative (i.e., a unital ∗-homomorphism) [35, Cor. 3.1.6]. Thus, the extreme pseudo-expectations
for (A, D) are precisely the multiplicative pseudo-expectations:
Ext PsExp(A, D) = PsExp× (A, D).
Theorem 3.19. Let (A, D) be an Abelian inclusion. Then the mapping
D-disjoint
given by Ψ → ker(Ψ)
PsExp× (A, D) → maximal
ideals of A
is a bijection. In particular, PsExp× (A, D) is a singleton iff there exists a
unique maximal D-disjoint ideal I A.
Proof. Let Ψ ∈ PsExp× (A, D). Then ker(Ψ) is a D-disjoint ideal of A,
and the map I → Ψ(I) is an order-preserving bijection between the Ddisjoint ideals of A containing ker(Ψ) and the D-disjoint ideals of Ψ(A).
Since (I(D), D) has the faithful unique pseudo-expectation property (Example 2.11), it is hereditarily essential, by Theorem 3.5. Thus (Ψ(A), D) is
essential, so that the only D-disjoint ideal of Ψ(A) is 0. It follows that the
only D-disjoint ideal of A containing ker(Ψ) is ker(Ψ) itself, which says that
ker(Ψ) is a maximal D-disjoint ideal of A.
Conversely, suppose I ⊆ A is a maximal D-disjoint ideal. The map
Ψ0 : I + D → D : h + d → d is a unital ∗-homomorphism. Since I(D) is an
injective unital Abelian C ∗ -algebra, there exists a unital ∗-homomorphism
Ψ : A → I(D) such that Ψ|I+D = Ψ0 [12]. Clearly, Ψ ∈ PsExp× (A, D) and
I ⊆ ker(Ψ) ⊆ A is a D-disjoint ideal. Thus, ker(Ψ) = I, by maximality.
Finally, suppose Ψ1 , Ψ2 ∈ PsExp× (A, D), with ker(Ψ1 ) = ker(Ψ2 ). Define
ι : Ψ1 (A) → Ψ2 (A) by the formula ι(Ψ1 (x)) = Ψ2 (x), x ∈ A. Then ι is a
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D. R. PITTS AND V. ZARIKIAN
unital ∗-isomorphism which fixes D. By injectivity, there exists an unital ∗homomorphism ι : I(D) → I(D) such that ι|Ψ1 (A) = ι. By the rigidity of the
injective envelope, ι = id, so that Ψ1 = Ψ2 .
Remark 3.20. Taking D = C in Theorem 3.19 above, one recovers the
well-known bijective correspondence between the characters and the maximal
ideals of A.
Corollary 3.21. Let (C(Y ), C(X)) be an Abelian inclusion with corresponding continuous surjection j : Y → X. The following four conditions are
equivalent:
(i) There exists a unique pseudo-expectation for (C(Y ), C(X)).
(ii) There exists a unique multiplicative pseudo-expectation for (C(Y ), C(X)).
(iii) There exists a unique maximal C(X)-disjoint ideal in C(Y ).
(iv) There exists a unique minimal closed set K ⊆ Y such that j(K) = X.
Proof. (i) ⇐⇒ (ii) follows from Theorem 3.18.
(ii) ⇐⇒ (iii) Theorem 3.19.
(iii) ⇐⇒ (iv) The map K →
{g ∈ C(Y ) : g|K = 0} is an order-reversing
bijection between the closed sets K ⊆ Y such that j(K) = X and the C(X)disjoint ideals in C(Y ).
Corollary 3.22. Let (C(Y ), C(X)) be an Abelian inclusion with corresponding continuous surjection j : Y → X. Then the following are equivalent:
(i) There exists a unique pseudo-expectation for (C(Y ), C(X)), which is
faithful.
(ii) There exists a unique multiplicative pseudo-expectation for (C(Y ), C(X)),
which is faithful.
(iii) (C(Y ), C(X)) is essential.
(iv) If K ⊆ Y is closed and j(K) = X, then K = Y .
Proof. Same proof as Corollary 3.21.
4. Examples
Now we provide additional examples of C ∗ -inclusions with (and without)
the unique pseudo-expectation property (resp. the faithful unique pseudoexpectation property). In proving that these examples are actually examples,
we will take advantage of some of the general theory developed so far.
In Section 2.1, we mentioned that the unique pseudo-expectation property is not hereditary from below. Equipped with the results of the previous
section, it is easy to give an example which demonstrates this.
Example 4.1. There exist Abelian inclusions D ⊆ D0 ⊆ A such that (A, D)
has the unique pseudo-expectation property, but (A, D0 ) does not. That is,
the unique pseudo-expectation property is not hereditary from below.
UNIQUE PSEUDO-EXPECTATIONS
467
Proof. Let X = [0, 1], X0 = [0, 1] ∪ {2}, and Y = [0, 1] ∪ {2, 3}. Define continuous surjections j : X0 → X and k : Y → X0 by the formulas
t, t ∈ [0, 1],
t, t ∈ [0, 1],
j(t) =
and k(t) =
1, t = 2
2, t ∈ {2, 3}.
Then i = j ◦ k : Y → X is the continuous surjection
t, t ∈ [0, 1],
i(t) =
1, t ∈ {2, 3}.
Clearly there exists a unique minimal closed set K ⊆ Y such that i(K) = X,
namely K = [0, 1]. On the other hand, there are multiple minimal closed
sets L ⊆ Y such that k(L) = X0 , for example both L = [0, 1] ∪ {2} and L =
[0, 1] ∪ {3}. Thus by Corollary 3.21, we have inclusions C(X) ⊆ C(X0 ) ⊆
C(Y ) such that (C(Y ), C(X)) has the unique pseudo-expectation property,
but (C(Y ), C(X0 )) does not.
We can also provide examples of the poor behavior of the unique pseudoexpectation property with respect to quotients described in Section 3.3. To
that end, let (C(Y ), C(X)) be an Abelian inclusion, with corresponding continuous surjection j : Y → X. Suppose Z ⊆ Y is closed and J = {g ∈ C(Y ) :
g|Z = 0} C(Y ). Then J ∩ C(X) = {f ∈ C(X) : f |j(Z) = 0} C(X). Thus
C(Y )/J , C(X)/ J ∩ C(X) ∼
= (C(Z), C j(Z) ,
with corresponding continuous surjection j|Z : Z → j(Z). Furthermore, J ∩
C(X) C(X) is regular iff j(Z)◦ = j(Z), where the interior and closure are
calculated in X.
Example 4.2. There exists an Abelian inclusion (A, D) and J A such
that (A, D) has the unique pseudo-expectation property, but (A/J , D/(J ∩
D)) does not. Of course, J ∩ D D is not a regular ideal.
Proof. Let Y = ([0, 1] × {0}) ∪ ({1} × [0, 1]) ⊆ [0, 1] × [0, 1], X = [0, 1], and
j : Y → X be defined by the formula j(s, t) = s, (s, t) ∈ Y . Then there exists a
unique minimal closed set K ⊆ Y such that j(K) = X, namely K = [0, 1]×{0}.
Thus, (C(Y ), C(X)) has the unique pseudo-expectation property, by Corollary 3.21. Now let Z = {1} × [0, 1], a closed subset of Y . Then j(Z) = {1}, and
there does not exist a unique minimal closed set L ⊆ Z such that j(L) = j(Z).
Thus, (C(Z), C(j(Z)) does not have the unique pseudo-expectation property.
Of course j(Z)◦ = ∅ j(Z).
Example 4.3. There exists an Abelian inclusion (A, D) and J A such
that (A, D) has the faithful unique pseudo-expectation property and J ∩ D is
a regular ideal in D, but (A/J , D/(J ∩ D)) does not have the faithful unique
pseudo-expectation property.
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D. R. PITTS AND V. ZARIKIAN
Proof. Let Y = ([0, 1/2] × {0}) ∪ ([1/2, 1] × {1}) ⊆ [0, 1] × [0, 1], X = [0, 1],
and j : Y → X be defined by the formula j(s, t) = s, (s, t) ∈ Y . Then there
exists a unique minimal closed set K ⊆ Y such that j(K) = X, namely K = Y .
Thus (C(Y ), C(X)) has the faithful unique pseudo-expectation property, by
Corollary 3.22. Now let Z = ([0, 1/2] × {0}) ∪ {(1/2, 1)}, a closed subset of Y .
Then j(Z) = [0, 1/2], so that j(Z)◦ = j(Z). There exists a unique minimal
closed set L ⊆ Z such that j(L) = j(Z), namely L = [0, 1/2] × {0}. Since
L Z, (C(Z), C(j(Z))) has the unique pseudo-expectation property, but not
the faithful unique pseudo-expectation property.
In the Introduction we mentioned that the inclusion (B(L2 [0, 1]), C[0, 1])
admits no conditional expectations (Example 1.3). An interesting question
(posed to us by Philip Gipson) is whether or not (B(L2 [0, 1]), C[0, 1]) has
a unique pseudo-expectation. It turns out that even the Abelian inclusion
(L∞ [0, 1], C[0, 1]) admits multiple pseudo-expectations. We found it difficult
to fit this example into the context of Corollary 3.21, so we utilize Theorem 3.16 instead.
Example 4.4. The Abelian inclusion (L∞ [0, 1], C[0, 1]) has infinitely many
pseudo-expectations, none of which are faithful.
Proof. Let B ∞ [0, 1] be the C ∗ -algebra of bounded complex-valued Borel
functions on [0, 1]. Let N [0, 1] B ∞ [0, 1] be the Lebesgue-null functions, so
that B ∞ [0, 1]/N [0, 1] = L∞ [0, 1]. Likewise, let M [0, 1] B ∞ [0, 1] be the meager functions, so that B ∞ [0, 1]/M [0, 1] = D[0, 1], the Dixmier algebra. Recall
that D[0, 1] = I(C[0, 1]) [12].
Now let A ⊆ [0, 1] be a Borel set such that both A and Ac are measure
dense, meaning that |V ∩ A| > 0 and |V ∩ Ac | > 0 for every open set V ⊆ [0, 1].
(Here | · | stands for Lebesgue measure.) One possible construction of A can
be found in [33].
A measure-theoretic argument shows that if f ∈ C[0, 1]sa and f + N [0, 1] ≤
χA + N [0, 1], then f ≤ 0. Likewise, if f ∈ C[0, 1]sa and f + N [0, 1] ≥ χA +
N [0, 1], then f ≥ 1.
It follows that
sup f + M [0, 1] : f ∈ C[0, 1]sa , f + N [0, 1] ≤ χA + N [0, 1] = 0 + M [0, 1]
D[0,1]
and
inf
D[0,1]
f + M [0, 1] : f ∈ C[0, 1]sa , f + N [0, 1] ≥ χA + N [0, 1] = 1 + M [0, 1].
By Theorem 3.16, (L∞ [0, 1], C[0, 1]) does not have the unique pseudo-expectation property.
It remains to show that no pseudo-expectation for (L∞ [0, 1], C[0, 1]) is faithful. By [12, Thm. 2.21], there are C ∗ -inclusions C[0, 1] ⊆ I(C[0, 1]) ⊆ L∞ [0, 1].
Suppose Φ is a faithful pseudo-expectation for (L∞ [0, 1], C[0, 1]). Then
UNIQUE PSEUDO-EXPECTATIONS
469
Φ : L∞ [0, 1] → I(C[0, 1]) is a ucp map such that Φ|C[0,1] = id. By the rigidity
of the injective envelope, Φ|I(C[0,1]) = id. Thus, Φ is a faithful conditional
expectation of L∞ [0, 1] onto I(C[0, 1]). It follows from [17, Lemma 1] that
D[0, 1] = I(C[0, 1]) is a W ∗ -algebra, contradicting [19, Exercise 5.7.21].
Transformation group C ∗ -algebras. Let Γ be a discrete group acting on
a compact Hausdorff space X by homeomorphisms, and let C(X) r Γ be
the corresponding reduced crossed product (see [5, Ch. 4] for more details).
In this section, we examine the unique pseudo-expectation property for the
inclusions (C(X) r Γ, C(X)) and (C(X) r Γ, C(X)c ). We
recall that elements of C(X) r Γ have formal series representations
t∈Γ at λt , where
at ∈ C(X) for all t ∈ Γ, and that there exists a faithful conditional expectation E : C(X) r Γ → C(X), namely
a t λt = a e .
E
t∈Γ
For s ∈ Γ, write Fs = {x ∈ X : sx = x} for the fixed points of s, and for
x ∈ X, let
H x := s ∈ Γ : x ∈ (Fs )◦ .
The condition that H x is Abelian for every x ∈ X is equivalent to the condition
that C(X)c is Abelian [28, Theorem 6.6]. The following result shows that
when either of these equivalent conditions hold, then (C(X) r Γ, C(X)c ) has
the faithful unique pseudo-expectation property.
Proposition 4.5 ([28, Theorem 6.10]). If C(X)c is Abelian, then
(C(X) r Γ, C(X)c ) has the faithful unique pseudo-expectation property. In
particular, this happens when Γ is Abelian.
In general, we do not know a characterization of when (C(X) r Γ, C(X)c )
has the faithful unique pseudo-expectation property, or even the unique pseudo-expectation property. However, we do have the following result for the
inclusion (C(X) r Γ, C(X)).
Theorem 4.6. The following are equivalent:
(i) (C(X) r Γ, C(X)) has the unique pseudo-expectation property.
(ii) (C(X) r Γ, C(X)) has the faithful unique pseudo-expectation property.
(iii) C(X)c = C(X) (i.e., C(X) is a MASA in C(X) r Γ).
(iv) The action of Γ on X is topologically free (i.e., (Ft )◦ = ∅ for all e = t ∈ Γ).
If in addition, C(X) r Γ = C(X) Γ (e.g. when Γ is amenable), conditions
(i) through (iv) are equivalent to,
(v) (C(X) r Γ, C(X)) is essential.
Proof. (i) =⇒ (ii) If (C(X)r Γ, C(X)) has the unique pseudo-expectation
property, then
PsExp C(X) r Γ, C(X) = {E},
470
D. R. PITTS AND V. ZARIKIAN
and E is faithful.
(ii) =⇒ (iii) Suppose (C(X) r Γ, C(X)) has the faithful unique pseudoexpectation property. By Corollary 3.14, C(X)c is Abelian. By Proposition 2.6, E|C(X)c is the unique pseudo-expectation for (C(X)c , C(X)), and so
E|C(X)c is multiplicative by Corollary 3.21. It follows that E|C(X)c = id, so
that C(X)c = C(X).
(iii) =⇒ (i) Since C(X)c = C(X), (C(X) r Γ, C(X)) is a regular MASA
inclusion. Thus (C(X) r Γ, C(X)) has the unique pseudo-expectation property, by Pitts’ Theorem 1.4.
(iii) ⇐⇒ (iv) By [28, Prop. 6.3]
c
◦
at λt ∈ C(X) r Γ : supp(at ) ⊆ (Ft ) , t ∈ Γ .
C(X) =
t∈Γ
Thus C(X)c = C(X) if and only if (Ft )◦ = ∅ for all e = t ∈ Γ, if and only if
the action of Γ on X is topologically free.
Finally, assume that C(X) Γ = C(X) r Γ. Theorem 3.5 shows that (ii)
implies (v), and [21, Thm. 4.1] gives (v) implies (iv).
5. W ∗ -inclusions
In this section, we investigate the unique pseudo-expectation property for
W ∗ -inclusions (M, D). This means that (M, D) is a C ∗ -inclusion such that
M is a W ∗ -algebra and D is σ(M, M∗ )-closed.
First we consider Abelian W ∗ -inclusions. Corollary 3.21 above shows that
there exist nontrivial Abelian C ∗ -inclusions (C(Y ), C(X)) with the unique
pseudo-expectation property. Not so for Abelian W ∗ -inclusions, due to the
abundance of normal states.
Theorem 5.1. Let (M, D) be a W ∗ -inclusion.
(i) Suppose M is Abelian. Then (M, D) has the unique pseudo-expectation
property iff M = D.
(ii) More generally, let (M, D) be a W ∗ -inclusion, with D Abelian (and M
possibly non-Abelian). If (M, D) has the unique pseudo-expectation property, then D is a MASA in M.
Observe that because D is an Abelian von Neumann algebra, the pseudoexpectations in the theorem are conditional expectations.
Proof. (i) Suppose PsExp(M, D) = {E}. Let a ∈ Msa . Since M is
Abelian,
{d ∈ Dsa : d ≤ a}
is an increasing net indexed by itself. Indeed, if f, g ≤ h are continuous functions, then max{f, g} ≤ h. By Theorem 3.16, we have that
E(a) = sup{d ∈ Dsa : d ≤ a}.
D
UNIQUE PSEUDO-EXPECTATIONS
471
Now let φ ∈ (D∗ )+ and φ ∈ (M∗ )+ be an extension. Then
φ E(a) = sup φ(d) : d ≤ a ,
by normality. On the other hand, if d ≤ a, then φ(d) = φ(d) ≤ φ(a), which
implies φ(E(a)) ≤ φ(a). Replacing a by −a, we conclude that φ(E(a)) = φ(a),
and so φ = φ ◦ E. Thus if a ∈ M and ψ ∈ (M∗ )+ , then ψ = ψ|D ◦ E, which
implies ψ(a) = ψ(E(a)). Since the choice of ψ was arbitrary, a = E(a) ∈ D.
(ii) Let D ⊆ A ⊆ M be a MASA. Since (M, D) has the unique pseudoexpectation property, so does (A, D), by Proposition 2.6. Then A = D, by (i)
above.
Our next objective is to generalize Theorem 5.1, by showing that for an arbitrary W ∗ -inclusion (M, D), the unique pseudo-expectation property implies
that Dc = Z(D) (Corollary 5.3). Our proof relies on a nice bijective correspondence between the conditional expectations Dc → Z(D) and the conditional
expectations C ∗ (D, Dc ) → D (Theorem 5.2). Theorem 5.2 is related to [8,
Thm. 5.3], but to our knowledge, is new. Our proof of Theorem 5.2 uses some
fairly recent technology, which we now describe.
Let M, N be W ∗ -algebras and Z ⊆ Z(M) ∩ Z(N ) be a W ∗ -subalgebra. By
[4], [11], there exist on the Z-balanced algebraic tensor product M ⊗Z N both
a minimal C ∗ -norm · min and a maximal C ∗ -norm · max , which coincide
if either M or N is Abelian. When Z = C, this fact is now classical, see [36,
Chapter IV.4]. Now suppose that for i = 1, 2, Mi , Ni are W ∗ -algebras and
Z ⊆ Z(Mi ) ∩ Z(Ni ) is a W ∗ -subalgebra. If u : M1 → M2 and v : N1 → N2 are
completely contractive Z-bimodule maps, then the unique Z-bimodule map
u ⊗ v : (M1 ⊗Z N1 , · min ) → (M2 ⊗Z N2 , · min ) such that (u ⊗ v)(x ⊗ y) =
u(x) ⊗ v(y) for all x ∈ M1 , y ∈ N1 is a contraction. Furthermore, if M1 ⊆ M2
and N1 ⊆ N2 , then (M1 ⊗Z N1 , · min ) ⊆ (M2 ⊗Z N2 , · min ).
Theorem 5.2. Let (M, D) be a W ∗ -inclusion. Then the map E → E|Dc is
a bijective correspondence between the conditional expectations C ∗ (D, Dc ) → D
and the conditional expectations Dc → Z(D). In particular, there exists a
conditional expectation C ∗ (D, Dc ) → D.
Proof. Let E : C ∗ (D, Dc ) → D be a conditional expectation. For all d ∈ Dc
and d ∈ D, we have that
dE d = E dd = E d d = E d d,
which implies E(d ) ∈ Z(D). Conversely, suppose θ : Dc → Z(D) is a conditional expectation. By [19, Thm. 5.5.4],
n
n
di ⊗ di →
di di
D ⊗Z(D) Dc → Alg D, Dc ⊆ M :
i=1
i=1
472
D. R. PITTS AND V. ZARIKIAN
is a ∗-isomorphism. Thus, (with the notation of the previous paragraph)
n
n
n
n
di ⊗ di ≤
di di ≤ di ⊗ di ,
di ⊗ di ∈ D ⊗Z(D) Dc .
i=1
min
i=1
i=1
max
i=1
Furthermore, since Z(D) is Abelian,
n
n
n
di ⊗ zi =
di zi ,
di ⊗ zi ∈ D ⊗Z(D) Z(D).
i=1
i=1
i=1
min
n
Thus for all i=1 di ⊗ di ∈ D ⊗Z(D) Dc ,
n
n
n
n
d i θ di = d i ⊗ θ di ≤
di ⊗ di ≤
di di .
i=1
i=1
min
i=1
min
i=1
It follows that the map
n
n
di di →
di θ di
Alg D, Dc → Alg D, Z(D) = D :
i=1
i=1
extends uniquely to a conditional expectation Θ : C ∗ (D, Dc ) → D. Clearly the
maps E → E|Dc and θ → Θ described above are inverse to one another.
Recall that in the case of a C ∗ -inclusion, the faithful unique pseudo-expectation property implies that Dc is Abelian, but we do not know whether the
faithfulness assumption can be dropped. However, the following corollary
to Theorem 5.2 shows that in the W ∗ -case, faithfulness is not necessary to
conclude Dc is Abelian. In fact, more is true.
Theorem 5.3. Let (M, D) be a W ∗ -inclusion. If (M, D) has the unique
pseudo-expectation property, then Dc = Z(D).
Proof. If (M, D) has the unique pseudo-expectation property, then so does
(C ∗ (D, Dc ), D), by Proposition 2.6. By Theorem 5.2, there exists a unique
conditional expectation C ∗ (D, Dc ) → D, therefore a unique conditional expectation Dc → Z(D). By Theorem 5.1, Z(D) ∩ Dc = Z(D). But
Z(D) ∩ Dc = Z(D) ∩ D ∩ M = D ∩ M = Dc .
We now turn to our main purpose in this section—characterizing the unique
pseudo-expectation property for various classes of W ∗ -inclusions (see Theorems 5.5 and 5.6).
The statement of Theorem 5.6 involves the tracial ultrapower construction,
which we recall for the reader. Let M be a II1 factor with trace τ , and let
ω ∈ βN \ N be a free ultrafilter. The tracial ultrapower of M with respect to
ω is defined to be Mω = ∞ (M)/Iω , where
Iω = (xn ) ∈ ∞ (M) : lim τ x∗n xn = 0 ∞ (M).
ω
UNIQUE PSEUDO-EXPECTATIONS
473
It can be shown that Mω itself is a II1 factor with trace
τω (xn ) + Iω = lim τ (xn ).
ω
The map M → Mω : x → (x) + Iω is an embedding. If D ⊆ M is a MASA,
then Dω = (∞ (D) + Iω )/Iω ⊆ Mω is a MASA. See [34, Appendix A] for more
details.
The proofs of Theorems 5.5 and 5.6 require some standard facts about
conditional expectations, which we collect into a proposition for the reader’s
convenience.
Proposition 5.4.
(i) Let I be an index set and for i ∈ I, let Mi ⊆ B(Hi ) be a W ∗ -algebra.
Then there exists a bijective correspondence between families of conditional
expectations{Ei : B(Hi ) → Mi }i∈I and conditional expectations
θ : B( i∈I Hi ) → i∈I Mi . Namely
θ [xij ] =
Ei (xii ).
i∈I
We have that θ is normal (resp. faithful) iff every Ei is normal (resp.
faithful).
(ii) For i = 1, 2, let (Mi , Di ) be a W ∗ -inclusion and Ei : Mi → Di be a
conditional expectation. Then there exists a conditional expectation
E : M1 ⊗ M2 → D1 ⊗ D2 such that E(x1 ⊗ x2 ) = E1 (x1 ) ⊗ E2 (x2 ) for
all x1 ∈ M1 , x2 ∈ M2 . If E1 and E2 are normal, then there exists a
unique normal conditional expectation E as above. [38, Thm. 4]
(iii) Let (M, D) be a W ∗ -inclusion. Then there exists a bijective correspondence between conditional expectations E : M → D and conditional expectations θ : B(K) ⊗ M → B(K) ⊗ D. Namely
θ [xij ] = E(xij ) .
We have that E is normal (resp. faithful) iff θ is normal (resp. faithful).
Now we come to the main results of this section. The first characterizes the
unique pseudo-expectation property for W ∗ -inclusions of the form (B(H), D),
and the second characterizes the unique pseudo-expectation property for W ∗ inclusions (M, D) when M∗ is separable and D is Abelian. In the latter
result, the separability hypothesis cannot be removed.
Theorem 5.5.
(i) Let A ⊆ B(H) be an Abelian W ∗ -algebra. Then (B(H), A) has the
unique pseudo-expectation property iff A is an atomic MASA. The unique
pseudo-expectation is a normal faithful conditional expectation.
474
D. R. PITTS AND V. ZARIKIAN
(ii) Generalizing (i), let M ⊆ B(H) be a W ∗ -algebra. Then (B(H), M) has
the unique pseudo-expectation property iff M is Abelian and atomic. The
unique pseudo-expectation is a normal faithful conditional expectation. In
particular, M is type I (and injective).
Proof. (i) By Theorem 5.1, we may assume that A ⊆ B(H) is a MASA. We
have the unitary equivalence
A = Aatomic ⊕ Adiffuse ,
2
where Aatomic is spatially isomorphic to ∞ (κ) acting
on ∞(κ) forαisome index set κ, and Adiffuse is spatially isomorphic to
i∈I L ([0, 1] ) acting
2
αi
on
L
([0,
1]
)
for
some
index
set
I
and
cardinals
αi , i ∈ I (see [25]).
i∈I
It is easy to see that for κ nonempty, there exists a unique conditional expectation B(2 (κ)) → ∞ (κ), which is normal and faithful. On the other
hand, for any nonzero cardinal α, there are multiple conditional expectations
B(L2 ([0, 1]α )) → L∞ ([0, 1]α ). Indeed, this is well known when α = 1 (Example 2.9), and follows from Proposition 5.4 (ii) and the unitary equivalence
L∞ [0, 1]α = L∞ [0, 1] ⊗ L∞ [0, 1]β ⊆ B L2 [0, 1] ⊗ B L2 [0, 1]β
when α > 1 (here β = α − 1 if α is finite, and β = α if α is infinite). The result
now follows from Proposition 5.4(i).
(ii) By Theorem 5.3, we may assume that M = Z(M), so that
H=
2m ⊗ Hm ,
M =
Im ⊗ Am , and M =
B 2m ⊗ Am ,
m
m
m
where Am ⊆ B(Hm ) is a MASA for each m. In particular, M is injective, so that pseudo-expectations for (B(H), M) are actually conditional
expectations. By Proposition 5.4(i) and (iii), there exists a unique conditional expectation B(H) → M iff there exist unique conditional expectations
B(Hm ) → Am for each m, iff Am is atomic for each m (by part (i) above).
The result now follows.
Theorem 5.6.
(i) Let (M, D) be a W ∗ -inclusion, with M∗ separable and D Abelian. Then
(M, D) has the unique pseudo-expectation property iff M is type I, D is a
MASA, and there exists
a family {pt } of Abelian projections for M such
that {pt } ⊆ D and t pt = 1. The unique pseudo-expectation is a normal
faithful conditional expectation.
(ii) Let (M, D) be a W ∗ -inclusion, with M a II1 factor and D a singular
MASA. If ω ∈ βN \ N, then (Mω , Dω ) has the faithful unique pseudoexpectation property. The unique pseudo-expectation is a normal, tracepreserving conditional expectation.
UNIQUE PSEUDO-EXPECTATIONS
475
Proof. (i) (⇒) Suppose there exists a unique expectation E : M → D. Then
D is a MASA, by Theorem 5.1. Since M∗ is separable, D is singly-generated.
Then E is normal and faithful, by [1, Cor. 3.3]. It follows that M is type I,
by [31, Thm. 3.3].
Thus there exist Abelian projections {pt } for M such that
{pt } ⊆ D and t pt = 1, by [1, Thm. 4.1].
(⇐) Conversely, if M is type I, D is a MASA,
and there exist Abelian
projections {pt } for M such that {pt } ⊆ D and t pt = 1, then there exists a
unique conditional expectation E : M → D, by [1, Thm. 4.1].
(ii) By [30, Thm. 0.1], (Mω , Dω ) has the unique extension property (UEP).
By Example 2.12, (Mω , Dω ) has the unique pseudo-expectation property,
and the unique pseudo-expectation is a conditional expectation, necessarily
normal, faithful, and trace-preserving.
Remark 5.7. Contrasting statements (i) and (ii) of Theorem 5.6 above, we
see that separability plays a role in the unique pseudo-expectation property.
6. Applications
In this section, we show that the faithful unique pseudo-expectation property can substantially simplify C ∗ -envelope calculations, and we relate the
faithful unique pseudo-expectation property to norming in the sense of Pop,
Sinclair, and Smith.
6.1. C ∗ -envelopes. Let C be a unital C ∗ -algebra and X ⊆ C be a unital
operator space such that C ∗ (X ) = C. There exists a unique maximal closed
two-sided ideal J C such that quotient map q : C → C/J is completely isometric on X [14]. Then Ce∗ (X ) = C/J is the C ∗ -envelope of X , the (essentially) unique minimal C ∗ -algebra generated by a completely isometric copy
of X . In general, determining Ce∗ (X ) can be quite challenging. However, if
D ⊆ X ⊆ C, and (C, D) has the faithful unique pseudo-expectation property,
then determining Ce∗ (X ) is not hard at all.
Theorem 6.1. Let (C, D) be a C ∗ -inclusion with the faithful unique pseudoexpectation property (more generally, such that every pseudo-expectation is
faithful). If D ⊆ X ⊆ C is an operator space, then Ce∗ (X ) = C ∗ (X ). That is,
the C ∗ -envelope equals the generated C ∗ -algebra.
Proof. By the previous discussion, Ce∗ (X ) = C ∗ (X )/J , where J C ∗ (X )
is the unique maximal closed two-sided ideal such that q : C ∗ (X ) → C ∗ (X )/J
is completely isometric on X . Since D ⊆ X , J must be D-disjoint. But then
J = 0, since (C, D) is hereditarily essential, by Theorem 3.5.
We say that a C ∗ -inclusion (C, D) is C ∗ -envelope determining if Ce∗ (X ) =
C (X ) for every operator space D ⊆ X ⊆ C. With this terminology, Theorem 6.1 becomes the implication
∗
every pseudo-expectation faithful =⇒ C ∗ -envelope determining.
476
D. R. PITTS AND V. ZARIKIAN
The converse is false.
Example 6.2. Let C = M2×2 (C) and D = CI. Then (C, D) is C ∗ -envelope
determining, but admits multiple pseudo-expectations, some of which are not
faithful.
Proof. Let C ⊆ X ⊆ M2×2 (C) be an operator space. Then dim(C ∗ (X )) ∈
{1, 2, 4}. If dim(C ∗ (X )) ∈ {1, 2}, then C ∗ (X ) = X , which implies Ce∗ (X ) = X .
Otherwise, if dim(C ∗ (X )) = 4, then C ∗ (X ) = M2×2 (C), which implies
Ce∗ (X ) = C ∗ (X ), since M2×2 (C) is simple.
6.2. Norming. According to Pitts’ Theorem 1.4, if (C, D) is a regular MASA
inclusion with the faithful unique pseudo-expectation property, then D norms
C in the sense of Pop, Sinclair, and Smith [29]. In this section, we investigate the relationship between the faithful unique pseudo-expectation property
and norming, for arbitrary C ∗ -inclusions. We show that the faithful unique
pseudo-expectation is conducive to norming (Theorem 6.8), but does not imply it (Example 6.9).
We begin by recalling the definition of norming, and proving some general
norming results which we will need later. Some of these results may be of
independent interest.
Definition 6.3. We say that an inclusion (C, D) is norming if for any
X ∈ Md×d (C), we have that
X = sup RXC : R ∈ Ball M1×d (D) , C ∈ Ball Md×1 (D) .
Proposition 6.4. Let (M, D) be a W ∗ -inclusion and {pt } ⊆ D be an increasing net of projections such that supt pt = 1. If (pt Mpt , pt Dpt ) is norming
for all t, then (M, D) is norming.
Proof. Let H be the Hilbert space on which M acts. Fix X ∈ Md×d (M)
and ε > 0. There exist ξ, η ∈ Ball(Hd ) such that
Xξ, η > X − ε.
Since supt pt = 1, there exists t such that
X(Id ⊗ pt )ξ, (Id ⊗ pt )η > Xξ, η − ε.
Set X̃ = (Id ⊗ pt )X(Id ⊗ pt ) ∈ Md×d (pt Mpt ). Since pt Dpt norms pt Mpt , there
exist R ∈ Ball(M1×d (pt Dpt )), C ∈ Ball(Md×1 (pt Dpt )) such that
RX̃C > X̃ − ε.
Then R ∈ Ball(M1×d (D)), C ∈ Ball(Md×1 (D)), and
RXC = R(Id ⊗ pt )X(Id ⊗ pt )C = RX̃C
> X̃ − ε ≥ X̃ξ, η − ε
UNIQUE PSEUDO-EXPECTATIONS
= X(Id ⊗ pt )ξ, (Id ⊗ pt )η − ε > Xξ, η − 2ε
> X − 3ε.
477
∗
, Di ) be a W -inclusion. If (Mi , Di ) is
Corollary 6.5. For i ∈ I, let (Mi
norming for all i ∈ I, then ( i∈I Mi , i∈I Di ) is norming.
Lemma 6.6. Let (C, D) be a C ∗ -inclusion and I C. If (D + I)/I norms
C/I, then for every X ∈ Md×d (C), there exist R ∈ Ball(M1×d (D)) and C ∈
Ball(Md×1 (D)) such that
RXC > X + Md×d (I) − ε.
Proof. Let π : C → C/I be the quotient map. Fix X ∈ Md×d (C) and
ε > 0. By assumption, there exist R ∈ M1×d (D) and C ∈ Md×1 (D) such that
π1×d (R) < 1, πd×1 (C) < 1, and
π1×d (R)πd×d (X)πd×1 (C) > πd×d (X) − ε.
Now the map
D/(D ∩ I) → (D + I)/I : d + D ∩ I → d + I
is a unital ∗-isomorphism, in particular a complete isometry. Thus
π1×d (R) = R + M1×d (I) = R + M1×d (D ∩ I).
It follows that there exists R̃ ∈ M1×d (D) such that R̃ < 1 and π1×d (R̃) =
π1×d (R). Likewise, there exists C̃ ∈ Md×1 (D) such that C̃ < 1 and
πd×1 (C̃) = πd×1 (C). Then
R̃X C̃ ≥ π1×d (R̃)πd×d (X)πd×1 (C̃) > πd×d (X) − ε.
Proposition 6.7. Let M be a II1 factor, D ⊆ M be a MASA, and ω ∈
βN \ N. If (Mω , Dω ) is norming, then so is (M, D).
Proof. By assumption, Dω = (∞ (D) + Iω )/Iω norms Mω = ∞ (M)/Iω .
Let X ∈ Md×d (M) and ε > 0. Then (X) ∈ Md×d (∞ (M)) = ∞ (Md×d (M)).
By Lemma 6.6, there exist (Rn ) ∈ M1×d (∞ (D)) = ∞ (M1×d (D)) and (Cn ) ∈
Md×1 (∞ (D)) = ∞ (Md×1 (D)) such that (Rn ) < 1, (Cn ) < 1, and
(Rn )(X)(Cn ) > (X) + Md×d (Iω ) − ε.
Thus,
sup Rn < 1,
n
sup Cn < 1,
n
and
sup Rn XCn > X − ε.
n
∗
Now we list some classes of C -inclusions for which (f!PsE) =⇒ (Norming).
Theorem 6.8. For the following classes of C ∗ -inclusions, the faithful
unique pseudo-expectation property implies norming:
(i) Regular MASA inclusions (C, D). In particular, C ∗ -inclusions (C(X) r
Γ, C(X)) for a discrete group Γ acting on a compact Hausdorff space X
by homeomorphisms.
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D. R. PITTS AND V. ZARIKIAN
(ii) Abelian inclusions (C(Y ), C(X)).
(iii) C ∗ -inclusions (C, D) with D ⊆ C ⊆ I(D) (i.e., operator space essential
inclusions).
(iv) W ∗ -inclusions (B(H), M).
(v) W ∗ -inclusions (M, D), with M∗ separable and D Abelian.
Proof. (i) This is Pitts’ Theorem 1.4.
(ii) Abelian inclusions (C(Y ), C(X)) are norming, whether or not they have
the faithful unique pseudo-expectation property [29, Ex. 2.5].
(iii) Let (C, D) be an operator space essential inclusion (see the discussion
preceding Example 2.11). For X ∈ Md×d (C), define
γd (X) = sup RXC : R ∈ Ball M1×d (D) , C ∈ Ball Md×1 (D) .
By [24, Thm. 2.1], γ = (γd )∞
d=1 is an operator space structure on C with the
following properties:
• γ1 (x) = x, x ∈ C;
• γd (X) ≤ X, X ∈ Md×d (C);
• γd (D) = D, D ∈ Md×d (D).
It follows that the identity map ι : (C, · ) → (C, γ) is a complete contraction
which is completely isometric on D. Since (C, D) is operator space essential,
ι is actually completely isometric on C, so (C, D) is norming.
(iv) By Theorem 5.6, M is Abelian. Let M ⊆ A ⊆ B(H) be a MASA.
Then A = A ⊆ M = M. Since A norms B(H), M norms B(H) as well.
(v) By Theorem 5.6, M is type I, D
is a MASA, and there exist
Abelian
projections {pt } ⊆ D for M such that t pt = 1. Letting pF = t∈F pt for
every finite set of indices F , we obtain an increasing net {pF } ⊆ D of finite
projections for M such that supF pF = 1. By Proposition 6.4, to prove that
(M, D) is norming it suffices to prove that (pF MpF , DpF ) is norming for each
F . Thus, we may assume that M is finite type I. By Corollary 6.5, we may
further assume that M is type In for some n ∈ N. There is a (non-spatial)
∗-isomorphism M = Mn×n (A) ⊆ B(Hn ), where A ⊆ B(H) is a MASA and
H is separable. By [18, Thm. 3.19], there exists a unitary u ∈ M such that
n
uDu∗ = ∞
n (A). It follows that D ⊆ B(H ) is a MASA, which implies that D
n
norms B(H ) [29, Thm. 2.7]. Thus, D norms M.
Example 6.9. There exists a II1 factor M and a singular MASA D ⊆ M
such that (M, D) has the faithful unique pseudo-expectation property, but D
does not norm M. Of course M∗ is non-separable.
Proof. Let F2 be the free group on two generators u and v, and let D =
W ∗ (u) ⊆ W ∗ (F2 ) = M. Then M is a II1 factor, D is a singular MASA, and
D does not norm M [29, Thm. 5.3]. Now let ω ∈ βN \ N. Then Mω is a
II1 factor, Dω is a singular MASA, and (Mω , Dω ) has the faithful unique
UNIQUE PSEUDO-EXPECTATIONS
479
pseudo-expectation property, by Theorem 5.6. But Dω does not norm Mω ,
by Proposition 6.7.
7. Conclusion
We conclude this paper with a list of questions and partial progress toward
some answers.
7.1. Questions.
Q1 Is the condition (Reg) hereditary from above? That is, if (C, D) is a
regular inclusion and D ⊆ C0 ⊆ C is a C ∗ -algebra, is (C0 , D) a regular
inclusion? (We expect that the answer is “no”.)
Q2 Is the condition (UEP) hereditary from below? That is, if (C, D) has the
unique extension property and D ⊆ D0 ⊆ C is a C ∗ -algebra, does (C, D0 )
have the unique extension property? (Again, we expect that the answer
is “no”.)
Q3 Is there a C ∗ -inclusion (C, D) with a unique conditional expectation, but
multiple pseudo-expectations?
Q4 If (C, D) has the unique extension property (UEP), does (C, D) have the
unique pseudo-expectation property? (By Example 2.12, the answer is
“yes” if D is Abelian.)
Q5 If (C, D) has the unique pseudo-expectation property, is Dc Abelian?
What if D is Abelian? (By Corollary 3.14, the answer is “yes” if (C, D)
has the faithful unique pseudo-expectation property.)
Q6 If every pseudo-expectation is faithful, is there a unique pseudoexpectation? Equivalently, by Theorem 3.5, does (C, D) have the faithful
unique pseudo-expectation property iff (C, D) is hereditarily essential?
Q7 Let Γ be a discrete group acting on a compact Hausdorff space X by
homeomorphisms. Find a condition on the action equivalent to (C(X)r
Γ, C(X)c ) having the unique pseudo-expectation property.
Q8 Is the C ∗ -inclusion (B(2 )/K(2 ), ∞ /c0 ) norming?
Q9 Is there a condition on a C ∗ -inclusion (C, D) which together with the
faithful unique pseudo-expectation property implies norming? In particular, is the separability of C such a condition?
Q10 Is there a condition on a W ∗ -inclusion (M, D) which together with the
faithful unique pseudo-expectation property implies norming? In particular, is the separability of M∗ such a condition? (By Theorem 6.8, the
answer is “yes” if D is Abelian.)
7.2. Progress on Questions 5 and 6.
Question 5. We are able to show that if (C, D) is a C ∗ -inclusion with D
Abelian, such that there exists a unique pseudo-expectation Φ, then Φ is multiplicative on Dc (Proposition 7.1). We regard this as partial progress toward
proving that Dc is Abelian. Indeed, by Corollary 3.21, if Dc is Abelian and Φ
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D. R. PITTS AND V. ZARIKIAN
is the unique pseudo-expectation for (C, D), then Φ necessarily is multiplicative on Dc .
Proposition 7.1. Let (C, D) be a C ∗ -inclusion, with D Abelian. If (C, D)
has unique pseudo-expectation Φ ∈ PsExp(C, D), then Φ is multiplicative
on Dc .
Proof. Let
MΦ = x ∈ C : Φ x∗ x = Φ(x)∗ Φ(x), Φ xx∗ = Φ(x)Φ(x)∗
be the multiplicative domain of Φ, the largest C ∗ -subalgebra of C on which
Φ is multiplicative [27, Thm. 3.18]. Suppose x ∈ (Dc )sa . Then Cx = C ∗ (D, x)
is a unital Abelian C ∗ -algebra containing D. By Proposition 2.6, (Cx , D) has
unique pseudo-expectation Φ|Cx , which is multiplicative by Corollary 3.21. It
follows that x ∈ MΦ .
Question 6. We show that if (M, D) is a W ∗ -inclusion with D injective, such
that every pseudo-expectation is faithful, then there exists a unique pseudoexpectation (Proposition 7.5).
Recall that a bounded linear map T between von Neumann algebras M
and N is singular if f ◦ T ∈ (M∗ )⊥ whenever f ∈ N∗ .
Lemma 7.2. Let (M, D) be a W ∗ -inclusion and θ : M → D be a completely
positive D-bimodule map. If θ is singular, then for every projection 0 = p ∈
Z(D), there exists a projection 0 = e ∈ M such that e ≤ p and θ(e) = 0.
Proof. Let {φi } ⊆ (D∗ )+ bea maximal family with mutually orthogonal
supports {s(φi )} ⊆ D. Then
i s(φi ) = 1, and so there exists j such that
s(φj )p = 0. Since θ is singular, φj ◦ θ ∈ (M∗ )⊥
+ . Thus there exists a projection
0 = e ∈ M such that e ≤ s(φj )p and φj (θ(e)) = 0 [36, Thm. III.3.8]. It follows
that s(φj )θ(e)s(φj ) = 0, which implies
θ(e) = θ s(φj )es(φj ) = s(φj )θ(e)s(φj ) = 0.
Lemma 7.3. Let (M, D) be a W ∗ -inclusion and θ : M → D be a completely
positive D-bimodule map. If CE(M, D) = ∅, then there exists E ∈ CE(M, D)
such that θ(x) = θ(1)E(x), x ∈ M. Furthermore, if 0 ≤ x ≤ s(θ(1)) and θ(x) =
0, then E(x) = 0.
Proof. Fix E0 ∈ CE(M, D). Let p = s(θ(1)) ∈ Z(D) and define
−1
E(x) = lim θ(1) + 1/k
θ(x) + p⊥ E0 (x), x ∈ M,
k→∞
where the limit exists in the strong operator topology (see [10, Lemma 5.1.6]).
Then E ∈ CE(M, D) and θ(x) = θ(1)E(x), x ∈ M. If 0 ≤ x ≤ p and θ(x) = 0,
then 0 ≤ E(x) ≤ p. But E(x) = p⊥ E0 (x), which implies E(x) = 0.
Lemma 7.4. Let (M, D) be a W ∗ -inclusion. If every conditional expectation M → D is faithful, then every conditional expectation M → D is normal.
UNIQUE PSEUDO-EXPECTATIONS
481
Proof. Let E : M → D be a conditional expectation. By [37], there exist
completely positive D-bimodule maps θn , θs : M → D such that θn is normal, θs is singular, and E = θn + θs . Assume that θs = 0. By Lemma 7.2,
there exists a projection 0 = e ∈ M such that e ≤ s(θs (1)) and θs (e) = 0.
By Lemma 7.3, there exists a conditional expectation Es : M → D such that
Es (e) = 0, a contradiction. Thus, E = θn is normal.
Proposition 7.5. Let (M, D) be a W ∗ -inclusion, with D injective. If every
pseudo-expectation Φ ∈ PsExp(M, D) is faithful, then (M, D) has the unique
pseudo-expectation property. In this situation, the unique pseudo-expectation
is a conditional expectation which is faithful and normal.
Proof. Since D is injective, PsExp(M, D) = CE(M, D). By Lemma 7.4,
every conditional expectation M → D is faithful and normal. By [8, Thm. 5.3],
the map E → E|Dc is a bijection between the faithful normal conditional
expectations M → D and the faithful normal conditional expectations Dc →
Z(D). Thus to show that there exists a unique conditional expectation M →
D, it suffices to show that there exists a unique faithful normal conditional
expectation Dc → Z(D). In fact, we will show that Dc = Z(D).
By Theorem 3.12, Dc is Abelian. Since every conditional expectation
M → D is faithful, every conditional expectation C ∗ (D, Dc ) → D is faithful, which implies every conditional expectation Dc → Z(D) is faithful, by
Theorem 5.2. In particular, every multiplicative conditional expectation
Dc → Z(D) is faithful, which implies Dc = Z(D).
References
[1] C. A. Akemann and D. Sherman, Conditional expectations onto maximal Abelian ∗subalgebras, J. Operator Theory 68 (2012), no. 2, 597–607. MR 2995737
[2] C. Akemann, S. Wassermann and N. Weaver, Pure states on free group C ∗ -algebras,
Glasg. Math. J. 52 (2010), no. 1, 151–154. MR 2587824
[3] R. J. Archbold, J. W. Bunce and K. D. Gregson, Extensions of states of C ∗ -algebras.
II, Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), no. 1–2, 113–122. MR 0667130
[4] E. Blanchard, Tensor products of C(X)-algebras over C(X). Recent advances in operator algebras (Orléans, 1992), Astérisque 232 (1995), 81–92. MR 1372526
[5] N. P. Brown and N. Ozawa, C ∗ -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI,
2008. MR 2391387
[6] L. J. Bunce and C.-H. Chu, Unique extension of pure states of C ∗ -algebras, J. Operator
Theory 39 (1998), no. 2, 319–338. MR 1620566
[7] M. D. Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly 90 (1983),
no. 5, 301–312. MR 0701570
[8] F. Combes and C. Delaroche, Groupe modulaire d’une espérance conditionnele dans
une algèbre de von Neumann, Bull. Soc. Math. France 103 (1975), no. 4, 385–426.
(French). MR 0415340
[9] R. S. Doran and A. V. Belfi, Characterizations of C ∗ -algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 101, Marcel Decker, Inc., New York,
NY, 1986. MR 0831427
482
D. R. PITTS AND V. ZARIKIAN
[10] E. G. Effros and Z.-J. Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York,
2000. MR 1793753
[11] T. Giordano and J. A. Mingo, Tensor products of C ∗ -algebras over Abelian subalgebras,
J. Lond. Math. Soc. (2) 55 (1997), no. 1, 170–180. MR 1423294
[12] D. Hadwin and V. I. Paulsen, Injectivity and projectivity in analysis and topology, Sci.
China Math. 54 (2011), no. 11, 2347–2359. MR 2859698
[13] M. Hamana, Injective envelopes of C ∗ -algebras, J. Math. Soc. Japan 31 (1979), no. 1,
181–197. MR 0519044
[14] M. Hamana, Injective envelopes of operator systems, Publ. Res. Inst. Math. Sci. 15
(1979), no. 3, 773–785. MR 0566081
[15] M. Hamana, Regular embeddings of C ∗ -algebras in monotone complete C ∗ -algebras,
J. Math. Soc. Japan 33 (1981), no. 1, 159–183. MR 0597486
[16] M. Hamana, The centre of the regular monotone completion of a C ∗ -algebra, J. Lond.
Math. Soc. (2) 26 (1982), no. 3, 522–530. MR 0684565
[17] R. V. Kadison, Operator algebras with a faithful weakly-closed representation, Ann. of
Math. (2) 64 (1956), 175–181. MR 0079233
[18] R. V. Kadison, Diagonalizing matrices, Amer. J. Math. 106 (1984), no. 6, 1451–1468.
MR 0765586
[19] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras.
Vol. I. Elementary theory, Pure and Applied Mathematics, vol. 100, Academic Press,
Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. MR 0719020
[20] R. V. Kadison and I. M. Singer, Extensions of pure states, Amer. J. Math. 81 (1959),
383–400. MR 0123922
[21] S. Kawamura and J. Tomiyama, Properties of topological dynamical systems and corresponding C ∗ -algebras, Tokyo J. Math. 13 (1990), no. 2, 251–257. MR 1088230
[22] A. Kumjian, On C ∗ -diagonals, Canad. J. Math. 38 (1986), no. 4, 969–1008.
MR 0854149
[23] S. S. Kutateladze, The Krein–Milman theorem and its converse, Sibirsk. Mat. Zh. 21
(1980), no. 1, 130–138. (Russian). MR 0562028
[24] B. Magajna, The minimal operator module of a Banach module, Proc. Edinb. Math.
Soc. (2) 42 (1999), no. 1, 191–208. MR 1669345
[25] D. Maharam, On homogeneous measure algebras, Proc. Natl. Acad. Sci. USA 28
(1942), 108–111. MR 0006595
[26] A. W. Marcus, D. A. Spielman and N. Srivastava, Interlacing families II: Mixed characteristic polynomials and the Kadison–Singer problem, Ann. of Math. (2) 182 (2015),
327–350. MR 3374963
[27] V. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies
in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002.
MR 1976867
[28] D. R. Pitts, Structure for regular inclusions, preprint, 2012; available at
arXiv:1202.6413v2 [math.OA].
[29] F. Pop, A. M. Sinclair and R. R. Smith, Norming C ∗ -algebras by C ∗ -subalgebras, J.
Funct. Anal. 175 (2000), no. 1, 168–196. MR 1774855
[30] S. Popa, A II1 factor approach to the Kadison–Singer problem, Comm. Math. Phys.
332 (2014), no. 1, 379–414. MR 3253706
[31] S. Popa and S. Vaes, Paving over arbitrary MASAs in von Neumann algebras, Anal.
PDE 8 (2015), no. 4, 1001–1023. MR 3366008
[32] J. Renault, Cartan subalgebras in C ∗ -algebras, Irish Math. Soc. Bull. 61 (2008), 29–63.
MR 2460017
[33] W. Rudin, Well-distributed measurable sets, Amer. Math. Monthly 90 (1983), no. 1,
41–42. MR 0691011
UNIQUE PSEUDO-EXPECTATIONS
483
[34] A. M. Sinclair and R. R. Smith, Finite von Neumann algebras and masas, London
Mathematical Society Lecture Note Series, vol. 351, Cambridge University Press, Cambridge, 2008. MR 2433341
[35] E. Stormer, Positive linear maps of operator algebras, Springer Monographs in Mathematics, Springer, Heidelberg, 2013. MR 3012443
[36] M. Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg,
1979. MR 0548728
[37] J. Tomiyama, On the projection of norm one in W ∗ -algebras. III, Tôhoku Math. J.
(2) 11 (1959), 125–129. MR 0107825
[38] J. Tomiyama, On the tensor products of von Neumann algebras, Pacific J. Math. 30
(1969), 263–270. MR 0246141
David R. Pitts, University of Nebraska–Lincoln, Lincoln, NE 68588, USA
E-mail address: [email protected]
Vrej Zarikian, U. S. Naval Academy, Annapolis, MD 21402, USA
E-mail address: [email protected]